Question: Express $z_1=0+10i$ in polar form. Express your answer in exact terms, using degrees, where your angle is between $0^\circ$ and $360^\circ$, inclusive. $z_1=$
Answer: The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $r$ For $z_1={0}+{10}i$ : ${a} = {0}$ ${b} = {10}$ Therefore, we can find ${r}$ as follows. $\begin{aligned}{r}&=\sqrt{{a}^2+{b}^2} \\\\&=\sqrt{{0}^2+{10}^2} \\\\&=\sqrt{0+100} \\\\&={\sqrt{100}} \\\\&={10}\end{aligned}$ Finding $\theta$ Since ${a}=0$ and ${b}$ is positive, ${\theta}$ must lie on the positive side of the imaginary number axis. Therefore its angle must be ${90^\circ}$. Summary $z_1={10}[\cos{90^\circ}+i\sin{90^\circ}]$